On the finiteness properties of local cohomology modules for regular ...

ON THE FINITENESS PROPERTIES OF LOCAL COHOMOLOGY MODULES FOR REGULAR LOCAL RINGS

arXiv:1703.00756v1 [math.AC] 2 Mar 2017

MONIREH SEDGHI, KAMAL BAHMANPOUR AND REZA NAGHIPOUR† Abstract. Let a denote an ideal in a regular local (Noetherian) ring R and let N be a finitely generated R-module with support in V (a). The purpose of this paper is to show that all homomorphic images of the R-modules ExtjR (N, Hai (R)) have only finitely many associated primes, for all i, j ≥ 0, whenever dim R ≤ 4 or dim R/a ≤ 3 and R contains a field. In addition, we show that if dim R = 5 and R contains a field, then the R-modules ExtjR (N, Hai (R)) have only finitely many associated primes, for all i, j ≥ 0.

1. Introduction In the present paper we continue the study of the finiteness properties of local cohomology modules for regular local rings. An interesting problem in commutative algebra is determining when the set of associated primes of the i-th local cohomology module Hai (R) of a Noetherian ring R with support in an ideal a of R, is finite. This question was raised by Huneke in [11] at the Sundance Conference in 1990. Examples given by A. Singh [25] (in the non-local case) and M. Katzman [16] (in the local case) show there exist local cohomology modules of Noetherian rings with infinitely many associated primes. However, in recent years there have been several results showing that this conjecture is true in many situations. The first result were obtained by Huneke and Sharp. In fact, Huneke and Sharp [12] (in the case of positive characteristic) have shown that, if R is a regular local ring containing a field, then Hai (R) has only finitely many associated primes for all i ≥ 0. Subsequently, G. Lyubeznik in [13] and [14] showed this result for unramified regular local rings of mixed characteristic and in characteristic zero. Further, Lyubeznik posed the following conjecture: Conjecture. If R is a regular ring and a an ideal of R, then the local cohomology modules Hai (R) have finitely many associated prime ideals for all i ≥ 0. While this conjecture remains open in this generality, several nice results are now available, see [3, 10, 19]. In lower dimensional cases, Marley in [19] showed that the set of associated prime ideals of the local cohomology modules Hai (M) is finite if R is a local ring Key words and phrases. Associated prime, cofinite module, local cohomology, minimax module, regular ring, weakly Laskerian module. 2010 Mathematics Subject Classification: 13D45, 14B15, 13H05. This research was in part supported by a grant from Azarbaijan Shahid Madani University (No. 217/d/917). † Corresponding author: e-mail: [email protected] (Reza Naghipour). 1

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M. SEDGHI, K. BAHMANPOUR AND R. NAGHIPOUR

of dimension d and a an ideal of R, M is a finitely generated R-module, in the following cases: (1) d ≤ 3; (2) d ≤ 4 and R has an isolated singularity; (3) d = 5 and R is an unramified regular local ring and M is torsion-free. For a survey of recent developments on finiteness properties of local cohomology modules, see Lyubezniks interesting paper [15]. The purpose of this paper is to provide some results concerning the set of associated primes of the local cohomology modules for a regular local ring, that almost results are extensions of Marley’s results on local cohomology modules over the strong ring (i.e. the regular local ring containing a field). Namely, we show that, for a finitely generated module N over a regular local ring R with support in V (a), the R-modules ExtjR (N, Hai (R)) are weakly Laskerian, for all i, j ≥ 0, whenever dim R ≤ 4 or dim R/a ≤ 3 and R contains a field. In addition, we show that if dim R = 5, then, for all i, j ≥ 0, the R-module ExtjR (N, Hai (R)) has only finitely many associated primes, when R contains a field. We say that an R-module M is said to be weakly Laskerian if the set of associated primes of any quotient module of M is finite [6]. Our main result in Section 2 is to establish some finiteness results of local cohomology modules for a regular local ring R with respect to an ideal a with dim R/a ≤ 3. More precisely, we prove the following. Theorem 1.1. Let R be a regular local ring containing a field with dim R = d ≥ 3, and let a be an ideal of R such that dim R/a ≤ 3. Then ExtjR (R/a, Hai (R)) is weakly Laskerian, for all i, j ≥ 0. The result of Theorem 1.1 is proved in Section 2. Pursuing this point of view further, we obtain the following consequence of Theorem 1.1, which is an extension of Marley’s result in [19]. Corollary 1.2. Let R be a regular local ring of dimension d ≤ 4 containing a field and a an ideal of R. Then, for any finitely generated R-module N with Supp(N) ⊆ V (a), the R-module ExtnR (N, Hai (R)) is weakly Laskerian, for all integers n, i ≥ 0. It will be shown in Section 3 that the subjects of Section 2 can be used to prove a finiteness result of local cohomology modules for a regular local ring of dimension 5. In fact, we will generalize the main result of Marley for a regular local ring of dimension 5. More precisely, we shall show that: Theorem 1.3. Let R be a regular local ring containing a field with dim R = 5 and let a be an ideal of R. Then the set AssR ExtnR (N, Hai (R)) is finite, for each finitely generated R-module N with support in V (a) and for all integers n, i ≥ 0. The proof of Theorem 1.3 is given in Section 3. The following proposition will be one of our main tools for proving Theorem 1.3. Proposition 1.4. Let R be a regular local ring of dimension d, and let a be an ideal of R with height a = 1. Then the set AssR ExtiR (N, Ha1(R)) is finite for each finitely generated R-module N with support in V (a) and for all integers i ≥ 0. As a consequence of Theorem 1.3 we derive the following result.

ON THE FINITENESS PROPERTIES OF LOCAL COHOMOLOGY MODULES

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Theorem 1.5. Let R be a regular local ring containing a field such that dim R ≤ 5, and let a be an ideal of R. Then, for all integers n, i ≥ 0 and any finitely generated R-module N with support in V (a), the set AssR ExtnR (N, Hai (R)) is finite. Finally, in this section we will show that, if M is a finitely generated module over a regular local ring R with dim R ≤ 4, then for any finitely generated R-module N with support in V (a), the R-module ExtnR (N, Hai (M)) is weakly Laskerian, for all integers i, n ≥ 0. In particular the set AssR Hai (M) is finite, for all integers i, n ≥ 0. Hartshorne [9] introduced the notion of a cofinite module, answering in negative a question of Grothendieck [8, Expos´ e XIII, Conjecture 1.1]. In fact, Grothendieck asked i if the modules HomR (R/a, Ha (M)) always are finitely generated for any ideal a of R and any finitely generated R-module M. This is the case when a = m, the maximal ideal in a local ring, since the modules Hmi (M) are Artinian. Hartshorne defined an R-module M to be a-cofinite if the support of M is contained in V (a) and ExtiR (R/a, M) is finitely generated for all i ≥ 0. In [26], H. Z¨oschinger, introduced an interesting class of minimax modules, and he has in [26, 27] given many equivalent conditions for a module to be minimax. An R-module N is said to be a minimax module, if there is a finitely generated submodule L of N, such that N/L is Artinian. The class of minimax modules thus includes all finitely generated and all Artinian modules. Also, an R-module M is called a-cominimax if the support of M is contained in V (a) and ExtiR (R/a, M) is minimax for all i ≥ 0. The concept of the a-cominimax modules is introduced in [2]. Throughout this paper, R will always be a commutative Noetherian ring with non-zero identity and a will be an ideal of R. For an R-module M, the i-th local cohomology module of M with respect to a is defined as Hai (M) = lim ExtiR (R/an , M). −→ n≥1

We refer the readers to [7] and [5] for more details on local cohomology. We shall use Min(a) to denote the set of all minimal primes of a. For each R-module L, we denote by AsshR L the set {p ∈ AssR L : dim R/p = dim L}. Also, for any ideal b of R, we denote {p ∈ Spec R : p ⊇ b} by V (b). Finally, for any ideal c of R, the radical of c, denoted by Rad(c), is defined to be the set {x ∈ R : xn ∈ c for some n ∈ N}. For any unexplained notation and terminology we refer the readers to [5] and [21]. 2. Finiteness of local cohomology modules for ideals of small dimension The purpose of this section is to study the finiteness properties of local cohomology modules for a regular local ring R with respect to an ideal a of R with dim R/a ≤ 3. The main goal is Theorem 2.7, which plays a key role in Section 3. This result extends a main result of T. Marley [19]. The following lemmas and proposition will be needed in the proof of Theorem 2.7. Theorem 2.1. (Huneke-Sharp and Lyubeznik). Let (R, m) be a regular local ring containing a field. Then for each ideal a of R and all integers i ≥ 0, the set of associated primes of the local cohomology modules Hai (R) are finite.

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Proof. See [12, 13, 14].

Lemma 2.2. Let (R, m) be a regular local ring containing a field with dim R = d ≥ 3. Let p1 , . . . , pn be prime ideals of R such that dim R/pi = 3 for all i = Tn1, . . . , n. Then d j Ha (R) = 0 and Supp(Ha (R)) is finite, for j = d − 1, d − 2, where a := i=1 pi . Proof. It follows easily from Lichtenbaum-Hartshorne and Grothendieck’s vanishing theorems that Had (R) = 0, Supp(Had−1 (R)) ⊆ {m} and for each q ∈ Supp(Had−2 (R)) we have dim R/q ≤ 1. Therefore dim Supp(Had−2 (R)) ≤ 1, and so Supp(Had−2 (R)) ⊆ AsshR Had−2 (R) ∪ {m}. Now it follows from Lemma 2.1 that Supp(Had−2 (R)) is a finite set, as required.

Corollary 2.3. Let (R, m) and a be as in Lemma 2.2. Then ExtiR (R/a, Haj (R)) is a weakly Laskerian R-module for all i ≥ 0 and for j = d − 1, d − 2. Proof. The result follows easily from Lemma 2.2.

The next lemma was proved by Melkersson for a-cofiniteness. The proof given in [22, Proposition 3.11] can be easily carried for weakly Laskerian modules. Lemma 2.4. Let R be a Noetherian ring, a an ideal of R, and M an R-module such that ExtiR (R/a, M) is a weakly Laskerian R-module for all i. If t is a non-negative integer such that the R-module ExtiR (R/a, Haj (M)) is weakly Laskerian, for all i and all j 6= t, then this is the case also when j = t. Proposition 2.5. Let (R, m) be a regular local ring containing a field with dim R = d ≥ 3. Let pT 1 , . . . , pn be prime ideals of R such that dim R/pi = 3, for all i = 1, . . . , n, and let a := ni=1 pi . Then ExtiR (R/a, Haj (R)) is a weakly Laskerian R-module for all i, j ≥ 0. Proof. If Hai (R) 6= 0, then it follows from Lemma 2.2 and [5, Theorems 6.1.2 and 6.2.7] that i ∈ {d −3, d −2, d −1}. Whence, the assertion follows from Corollary 2.3 and Lemma 2.4. The next result was proved by Kawasaki for finitely generated modules. The proof given in [17, Lemma 1] can be easily carried for weakly Laskerian modules. Lemma 2.6. Let R be a Noetherian ring, T an R-module, and a an ideal of R. Then the following conditions are equivalent: (i) ExtnR (R/a, T ) is weakly Laskerian for all n ≥ 0. (ii) For any finitely generated R-module N with support in V (a), ExtnR (N, T ) is weakly Laskerian for all n ≥ 0. We now are prepared to prove the main theorem of this section, which shows that when R is a regular local ring contains a field and a an ideal of R such that dim R/a ≤ 3, then all homomorphic images of the R-modules ExtjR (R/a, Hai (R)) have only finitely many associated primes. Theorem 2.7. Let (R, m) be a regular local ring containing a field with dim R = d ≥ 3, and let a be an ideal of R such that dim R/a ≤ 3. Then the R-module ExtjR (R/a, Hai (R)) is weakly Laskerian, for all i, j ≥ 0.


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Proof. In view of [4, Corollaries 2.7 and 3.2], we may assume that dim R/a = 3. Then we have grade a = d − 3. Hence, by virtue of [5, Theorems 6.1.2 and 6.2.7], Hai (R) = 0, whenever i 6∈ {d−3, d−2, d−1}. Moreover, by [23, Corollary 3.3], the set Supp(Had−1 (R)) is finite, in view of Proposition 2.5, it is enough for us to show that the R-module ExtjR (R/a, Had−3(R)) is weakly Laskerian. To this end, let X := {p ∈ Min(a) | height p = d − 3} and Y := {p ∈ Min(a) | height p ≥ d − 2}. T Then X 6= ∅. Set b = p∈X p. If Y = ∅, then Rad(a) = b, and so the assertion follows T from Proposition 2.5. Therefore we may assume that Y 6= ∅. Let c = p∈Y p. Then Rad(a) = b ∩ c and height c ≥ d − 2. Next, we show that height (b + c) ≥ d − 1. Suppose the contrary is true. Then there exists a prime ideal q of R such that b + c ⊆ q and height q = d − 2. Therefore there exist p1 ∈ X and p2 ∈ Y such that p1 + p2 ⊆ q. Since height p2 ≥ d − 2, it follows that q = p2 , and so p1 & p2 , which is a contradiction (note that p1 , p2 ∈ Min(a)). Consequently, grade (b + c) ≥ d − 1 and grade c ≥ d − 2. Therefore, it follows from [5, Theorem 6.2.7], that d−3 d−2 Hb+c (R) = 0 = Hb+c (R)

and

Hcd−3 (R) = 0.

It now follows from Rad(a) = b ∩ c and the Mayer-Vietoris sequence (see [5, Theorem 3.2.3]), that Had−3 (R) ∼ = Hbd−3 (R), and so by Proposition 2.5, the R-module j ExtR (R/b, Had−3(R)) is weakly Laskerian, for all j ≥ 0. On the other hand, since dim R/(b + c) ≤ 1, it is easy to see that V (b + c) = AsshR R/(b + c) ∪ {m}. Now, as Supp(ExtiR (R/c, Hbd−3(R))) ⊆ V (b + c), it follows that ExtiR (R/c, Hbd−3(R)) is a weakly Laskerian R-module, for all i ≥ 0. Also, as Supp(ExtiR (R/b + c, Hbd−3 (R))) ⊆ V (b + c), analogous to the preceding, we see that the R-module ExtiR (R/b + c, Hbd−3(R)) is also weakly Laskerian for all i ≥ 0. Now, the exact sequence 0 −→ R/ Rad(a) −→ R/b ⊕ R/c −→ R/b + c −→ 0, induces the long exact sequence · · · −→ ExtiR (R/b, Had−3 (R)) ⊕ ExtiR (R/c, Had−3(R)) −→ ExtiR (R/ Rad(a), Had−3 (R)) −→ ExtiR (R/b + c, Had−3(R)) −→ · · · , which shows that the R-module ExtiR (R/ Rad(a), Had−3 (R)) is weakly Laskerian, for all i ≥ 0, and so it follows from Lemma 2.6 that, the R-module ExtiR (R/a, Had−3(R)) is weakly Laskerian, for all i ≥ 0, as required.

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Corollary 2.8. Let (R, m) be a regular local ring of dimension d ≥ 3 containing a field, and let a be an ideal of R such that dim R/a ≤ 3. Then, for any finitely generated Rmodule N with Supp(N) ⊆ V (a), the R-module ExtnR (N, Hai (R)) is weakly Laskerian, for all integers n, i ≥ 0. Proof. The result follows from Theorem 2.7 and Lemma 2.6.

Corollary 2.9. Let (R, m) be a regular local ring of dimension d ≤ 4 containing a field, and a an ideal of R. Then, for any finitely generated R-module N with Supp(N) ⊆ V (a), the R-module ExtnR (N, Hai (R)) is weakly Laskerian, for all integers n, i ≥ 0. Proof. Since R is regular local, so dim R/a = 4 if and only if a = 0. Thus, the assertion is clear in the case of dim R/a = 4. Moreover, the case dim R/a = 3 follows from Corollary 2.8. Also, the case dim R/a = 2 follows from [4, Theorem 3.1] and Lemma 2.4. Finally, if dim R/a ≤ 1, then the assertion follows from [4, Theorem 2.6] and [17, Lemma 1]. 3. Finiteness of local cohomology modules for regular local rings of small dimension It will be shown in this section that the subjects of the previous section can be used to prove the finiteness of local cohomology modules for a regular local ring R with dim R = 5. The main result is Theorem 3.4. The following proposition will serve to shorten the proof of that theorem. The following easy lemma will be used in Proposition 3.2. f

g

Lemma 3.1. Let R be a Noetherian ring and let M ′ −→ M −→ M ′′ be an exact sequence of R-modules such that M ′ is weakly Laskerian and M ′′ has only finitely many associated primes. Then M has only finitely many associated primes. Proof. The assertion follows from the exact sequence 0 −→ M ′ /Im g −→ M −→ Im g −→ 0, by applying [21, Theorem 6.3].

Proposition 3.2. Let (R, m) be a d-dimensional regular local ring, and a an ideal of R such that height a = 1. Then the set AssR ExtiR (N, Ha1 (R)) is finite, for each finitely generated R-module N with support in V (a) and for all integers i ≥ 0. Proof. Let X := {p ∈ Min(a) | height p = 1} and Y := {p ∈ Min(a) | height p ≥ 2}. T Then X 6= ∅. Let b = p∈X p. Since R is a UFD, it follows from [21, Exercise 20.3] that b is a principal ideal, and so there is an element x ∈ R such that b = Rx. Hence, if Y = ∅, then Rad(a) = Rx, and so it follows from [18, Theorem 1] that, the R-module ExtiR (N, Ha1(R)) is finitely generated. Thus the set AssR ExtiR (N, Ha1 (R)) is T finite. Therefore, we may assume that Y 6= ∅. Then Rad(a) = Rx ∩ c, where c = p∈Y p. It is easy to see that height c ≥ 2 and height (b + c) ≥ 3. Whence, by using the Mayer-Vietoris sequence it yields that Ha1 (R) ∼ = Hb1 (R). Therefore, by [18, Theorem 1], the R-module 1 Ha (R) is b-cofinite.


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On the other hand, according to Artin-Rees lemma, there exists a positive integer n such that bn N ∩ Γb (N) = 0 = cn N ∩ Γc (N). We claim that bn N ∩ cn N = 0. To this end, suppose that q ∈ AssR (bn N ∩ cn N). Then q = (0 :R y), for some y(6= 0) ∈ bn N ∩ cn N. As Supp(N) ⊆ V (a), it follows that a ⊆ q, and so b ∩ c ⊆ q. Thus b ⊆ q or c ⊆ q, and so y ∈ Γb (N) or y ∈ Γc (N). Furthermore, since y ∈ bn N ∩ cn N, it follows that y = 0, which is a contradiction. Now, since bn N ∩ cn N = 0, the exact sequence 0 −→ N −→ N/bn N ⊕ N/cn N −→ N/(bn + cn )N −→ 0, induces the long exact sequence · · · −→ ExtiR (N/bn N, Ha1(R)) ⊕ ExtiR (N/cn N, Ha1 (R)) −→ ExtiR (N, Ha1 (R)) n n 1 −→ Exti+1 R (N/(b + c )N, Ha (R)) −→ · · · . Since Ha1 (R) is b-cofinite and

(†)

Supp(N/(bn + cn )N) ⊆ Supp(N/bn N) ⊆ V (b), it follows from [17, Lemma 1] that the R-modules ExtiR (N/bn N, Ha1 (R)) and ExtiR (N/(bn + cn )N, Ha1(R)) are finitely generated for all i ≥ 0. Next, let T := N/cn N and we show that ExtiR (T, Ha1 (R)) is also b-cofinite for all i ≥ 0. To do this, since Supp(Γb (T )) ⊆ V (b) it follows from [17, Lemma 1] that the R-module ExtiR (Γb (T ), Ha1(R)) is finitely generated for each i ≥ 0. Hence it is enough to show that the R-module ExtiR (T /Γb (T ), Ha1(R)) is finitely generated. As T /Γb (T ) is b-torsion-free, we therefore make the additional assumption that T is a b-torsion-free R-module. Then in view of [5, Lemma 2.11], x is a non-zerodivisor on T , and so the exact sequence x

0 −→ T −→ T −→ T /xT −→ 0 induces the long exact sequence x

· · · −→ ExtiR (T /xT, Ha1 (R)) −→ ExtiR (T, Ha1 (R)) −→ ExtiR (T, Ha1 (R)) 1 −→ Exti+1 R (T /xT, Ha (R)) −→ · · · .

(††)

Since Supp(T /xT ) ⊆ V (b) it follows from [17, Lemma 1] that ExtiR (T /xT, Ha1 (R)) is a finitely generated R-module, for all i ≥ 0, (note that b = Rx). Consequently, it follows from the exact sequence (††) that the R-modules (0 :ExtiR (T,Ha1 (R)) x) and ExtiR (T, Ha1(R))/xExtiR (T, Ha1 (R)) are finitely generated and hence b-cofinite. Therefore it follows from Melkersson’s result [22, Corollary 3.4] that the R-module ExtiR (T, Ha1 (R)) is b-cofinite for all i ≥ 0. Now, let Ω := ExtiR (N/bn N, Ha1 (R)) ⊕ ExtiR (N/cn N, Ha1 (R)). Then for every finitely generated submodule L of Ω the R-module Ω/L is also b-cofinite, and so the set AssR Ω/L is finite. Now, it follows from the exact sequence (†) and Lemma 3.1 that the set

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AssR ExtiR (N, Ha1 (R)) is finite, as required.

The next lemma was proved in [1] in the case R is local. The proof given in [1, Lemma 2.5] can be easily carried over Noetherian rings, so that we omit the proof. Lemma 3.3. Let R be a Noetherian ring, x an element of R and a an ideal of R such that a ⊆ Rad(Rx). Then, for any finitely generated R-module M, the R-homomorphism x Haj (M) −→ Haj (M) is an isomorphism, for each j ≥ 2. We are now ready to state and prove the main theorem of this section. Theorem 3.4. Let (R, m) be a five-dimensional regular local ring containing a field and a an ideal of R. Then the set AssR ExtnR (N, Hai (R)) is finite, for each finitely generated R-module N with support in V (a) and for all integers n, i ≥ 0. Proof. Since dim R/a = 5 if and only if a = 0, the assertion is clear in this case. Hence we consider the case when dim R/a ≤ 4. If dim R/a ≤ 3, then the result follows from the proof of Corollary 2.8. Therefore, we may assume that dim R/a = 4. Then height(a) = 1 and in view of the Lichtenbaum-Hartshorne vanishing theorem Ha5 (R) = 0. Whence Hai (R) = 0 whenever i 6∈ {1, 2, 3, 4}. Also, since by [23, Corollary 3.3], the set Supp(Ha4 (R)) is finite, it follows from Supp(ExtnR (N, Ha4 (R))) ⊆ Supp(Ha4 (R)) that the set AssR ExtnR (N, Ha4 (R)) is also finite. Consequently, in view of Proposition 3.2, we may consider the cases i = 2, 3. Case 1. i = 2. Suppose that X := {p ∈ Min(a) | height p = 1} and Y := {p ∈ Min(a) | height p ≥ 2}. T T Then X 6= ∅ and Y 6= ∅. Let b = p∈X p and c = p∈Y p. Since R is a UFD, it follows from [21, Ex. 20.3] that b is a principal ideal, and so there is an element x ∈ R such that b = Rx. Moreover, Rad(a) = Rx ∩ c, height c ≥ 2, and in view of Proposition 3.2, 1 1 ∼ 1 module. As we have S Ha (R) = Hb (R). Thus, by [18, Theorem 1], Ha (R) is a b-cofinite S c 6⊆ p∈X p, it follows that there is an element y ∈ c such that y 6∈ p∈X p. Now, in view of [24, Corollary 3.5], there exists the exact sequence 1 1 2 0 2 0 −→ HRy (HRx (R)) −→ HRx+Ry (R) −→ HRy (HRx (R)) −→ 0, 2 1 1 2 and so HRx+Ry (R) ∼ (HRx (R)), (note that HRx (R) = 0). = HRy Using again [24, Corollary 3.5], to show that there exists the exact sequence 1 2 0 0 −→ HRy (Ha1 (R)) −→ Ha+Ry (R) −→ HRy (Ha2 (R)) −→ 0, (†) and so it follows from Ha1 (R) ∼ = Hb1 (R) that H 1 (H 1 (R)) ∼ (R). = H 1 (H 1(R)) = H 1 (H 1 (R)) ∼ = H2 Ry

a

Ry

b

Ry

Rx

Also, since xy ∈ Rad(a), it follows that the R-module using Lemma 3.3, it is easy to see that the R-module

Rx+Ry

Ha2 (R) Ha2 (R)

is R(yx)-torsion. Hence is Ry-torsion. That is


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0 HRy (Ha2 (R)) = Ha2 (R). Consequently, from the exact sequence (†) we get the following exact sequence, 2 2 0 −→ HRx+Ry (R) −→ Ha+Ry (R) −→ Ha2 (R) −→ 0.

(††)

Furthermore, in view of Proposition 3.2, there exists a positive integer n such that the sequence 0 −→ N −→ N/bn N ⊕ N/cn N −→ N/(bn + cn )N −→ 0 is exact, and so we obtain the long exact sequence · · · −→ ExtiR (N/bn N, Ha2(R)) ⊕ ExtiR (N/cn N, Ha2 (R)) −→ ExtiR (N, Ha2 (R)) n n 2 −→ Exti+1 R (N/(b + c )N, Ha (R)) −→ · · · .

(†††)

Since Supp(N/(bn + cn )N) ⊆ Supp(N/bn N) ⊆ V (b), it follows from Lemma 3.3 that ExtjR (N/bn N, Ha2 (R)) = 0 = ExtjR (N/(bn + cn )N, Ha2 (R)), for all j ≥ 0. Whence, the exact sequence (†††) implies that j Extj (N, H 2(R)) ∼ = Ext (N/cn N, H 2(R)). R

a

R

a

Next, it is easy to see that height(Rx + Ry) = 2, and so height(a + Ry) = 2. Thus dim R/(a + Ry) = 3, and hence as Supp(N/cn N) ⊆ V (c) ⊆ V (a + Ry), 2 it follows from Corollary 2.8 that the R-module ExtjR (N/cn N, Ha+Ry (R)) is weakly Laskerian, for all j ≥ 0. Also, as grade(Rx + Ry) = 2, it follows from [5, Theorem 3.3.1] and [22, Proposition 2 3.11] that, the R-module HRx+Ry (R) is Rx+Ry-cofinite. Now, by modifying the argument 2 of the proof of Proposition 3.2, one can see that the R-module ExtjR (N/cn N, HRx+Ry (R)) j n 2 is Rx-cofinite for all j ≥ 0. In particular, AssR ExtR (N/c N, HRx+Ry (R)) is a finite set, for all j ≥ 0. Moreover, from the exact sequence (††), we deduce the long exact sequence 2 · · · −→ ExtjR (N/cn N, Ha+Ry (R)) −→ ExtjR (N/cn N, Ha2 (R)) n 2 −→ Extj+1 R (N/c N, HRx+Ry (R)) −→ · · · . Now using Lemma 3.1 and the above long exact sequence induced, it follows from the isomorphism j ExtjR (N, Ha2(R)) ∼ = ExtR (N/cn N, Ha2(R)),

that the set AssR ExtjR (N, Ha2(R)) is finite. Case 2. i = 3. Let X 6= ∅, Y 6= ∅ and y be as in the case 1. Then using the same argument, it follows from Lemma 3.3 that the R-modules Ha2 (R) and Ha3 (R) are Ry-torsion. Thus 0 1 (Ha3 (R)) = Ha3 (R). HRy (Ha2 (R)) = 0 and HRy

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Therefore, using the exact sequence 1 3 0 0 −→ HRy (Ha2 (R)) −→ Ha+Ry (R) −→ HRy (Ha3 (R)) −→ 0,

(†)

3 (see [24, Corollary 3.5]), we obtain that Ha3(R) ∼ (R). = Ha+Ry Moreover, it follows easily from the exact sequence

0 −→ N −→ N/bn N ⊕ N/cn N −→ N/(bn + cn )N −→ 0, that j ExtjR (N, Ha3(R)) ∼ = ExtR (N/cn N, Ha3(R)), 3 for all j ≥ 0. Hence, for all j ≥ 0 the R-module ExtjR (N/cn N, Ha+Ry (R)) is weakly j n 3 Laskerian, and so ExtR (N/c N, Ha (R)) is also a weakly Laskerian R-module. Therefore ExtjR (N, Ha3(R)) is a weakly Laskerian R-module, and hence it has finitely many associated primes, as required.

Theorem 3.5. Let (R, m) be a regular local ring containing a field such that dim R ≤ 5, and let a be an ideal of R. Then, for all integers n, i ≥ 0 and any finitely generated R-module N with support in V (a), the set AssR ExtnR (N, Hai (R)) is finite. Proof. The assertion follows from Corollary 2.8 and Theorem 3.4.

The final theorem of this section shows that if R is a regular local ring with dim R ≤ 4, then the R-module Hai (M) has finitely many associated primes, for all i ≥ 0 and for any finitely generated M over R. Recall that, an R-module L is called an a-cominimax module [2] if Supp(L) ⊆ V (a) and ExtiR (R/a, L) is minimax, for all i ≥ 0. Theorem 3.6. Let (R, m) be a regular local ring such that dim R ≤ 4. Suppose that a is an ideal of R and M a finitely generated R-module. Then for all integers i, j ≥ 0, the R-module Extj (R/a, Hai (M)) is weakly Laskerian. Proof. If dim R ≤ 3, then by virtue of [1, Theorem 2.12], the R-modules Hai (M) are acominimax, and so the R-module Extj (R/a, Hai (M)) is weakly Laskerian. Hence we may assume that dim R = 4. Now if dim R/a = 4, then a = 0 and so the result holds. Also, case dim R/a ≤ 2 follows from [4, Corollary 3.2]. Hence we may assume that dim R/a = 3. Then we have heigth(a) = 1. On the other hand, if dim M ≤ 2 then in view of [4, Corollary 2.7] and [22, Proposition 5.1], the R-module Hai (M) is a-cofinite, and so the result is clear. Also, in the case of dim M = 3, the assertion follows from [22, Proposition 5.1], [23, Corollary 3.3], the Grothendieck Vanishing Theorem and Lemma 2.4. Therefore we may assume that dim M = 4. Then in view of [22, Proposition 5.1], [23, Corollary 3.3], the Grothendieck Vanishing Theorem and Lemma 2.4, it is enough to show that the R-module Extj (R/a, Ha2(M)) is weakly Laskerian, for all j ≥ 0. To this end, let X := {p ∈ Min(a) | height p = 1} and Y := {p ∈ Min(a) | height p ≥ 2}.


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T Then X 6= ∅. Let b = p∈X p. If Y = ∅ then Rad(a) = b, and so Ha2 (M) = 0, as required. Therefore we may assume that Y 6= ∅. ThenTRad(a) = b ∩ c, height c ≥ 2, and in view of Theorem 2.7, height (b + c) ≥ 3, where c = p∈Y p. Now, since Rad(a) = b ∩ c, in view of the Mayer-Vietoris sequence the sequence f

g

2 3 Hb+c (M) −→ Hc2 (M) −→ Ha2 (M) −→ Hb+c (M)

(†)

is exact. Since dim R/(b + c) ≤ 1 we have V (b + c) = Min(b + c), and so as i Supp(Hb+c (M)) ⊆ V (b + c), i it follows that the set Supp(Hb+c (M)) is finite, for i = 2, 3. Therefore, it follows from the exact sequence (†) that the R-modules Ker f and Im g are weakly Laskerian. Thus, as dim R/c ≤ 2 it follows from [4, Corollary 3.3] that the R-module ExtiR (R/c, Hc2 (M)/Ker f ) is weakly Laskerian for all i ≥ 0. Now, the exact sequence

0 −→ Hc2 (M)/Ker f −→ Hc2 (M) −→ Im g −→ 0, induces the long exact sequence ExtiR (R/c, Hc2 (M)/Ker f ) −→ ExtiR (R/c, Hc2(M)) −→ ExtiR (R/c, Im g), for all i ≥ 0. Consequently, in view of Lemma 2.12, ExtiR (R/c, Hc2(M)) is a weakly Laskerian R-module. Finally, by using the exact sequence 0 −→ R/ Rad(a) −→ R/b ⊕ R/c −→ R/b + c −→ 0, we get the long exact sequence · · · −→ ExtiR (R/b + c, Ha2 (M)) −→ ExtiR (R/b ⊕ R/c, Ha2(M)) 2 −→ ExtiR (R/ Rad(a), Ha2(M)) −→ Exti+1 R (R/b + c, Ha (M)) −→ · · · . Now, by applying Lemma 2.10, we obtain that

ExtiR (R/b + c, Ha2(M)) = 0 = ExtiR (R/b, Ha2(M)), for each i ≥ 0, and therefore ExtiR (R/ Rad(a), Ha2(M)) ∼ = ExtiR (R/c, Ha2 (M)), for each i ≥ 0. Hence, the R-module ExtiR (R/ Rad(a), Ha2 (M)) is weakly Laskerian, for each i ≥ 0. Thus, it follows from Lemma 2.6 that, the R-module ExtiR (R/a, Ha2(M)) is also weakly Laskerian, for all i ≥ 0, and this completes the proof. We end this section with a result which is a generalization of Corollary 2.8. Corollary 3.7. Let the situation be as in Theorem 3.6. Then for any finitely generated R-module N with support in V (a), the R-module ExtjR (N, Hai (M)) is weakly Laskerian, for all integers i, j ≥ 0. In particular the set AssR ExtjR (N, Hai (M)) is finite, for all integers i, j ≥ 0. Proof. The assertion follows from Theorem 3.6 and Lemma 2.6.

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Acknowledgments The authors are deeply grateful to the referee for a very careful reading of the manuscript and many valuable suggestions in improving the quality of the paper. We also would like to thank from the Azarbaijan Shahid Madani University for the financial support (No. 217/d/917). References [1] N. Abazari and K. Bahmanpour, On the finiteness of Bass numbers of local cohomology modules, J. Alg. Appl. 10 (2011), 783-791. [2] J. A’zami, R. Naghipour and B. Vakili, Finiteness properties of local cohomology modules for aminimax modules, Proc. Amer. Math. Soc. 137 (2009), 439-448. [3] K. Bahmanpour and R. Naghipour, Associated primes of local cohomology modules and Matlis duality, J. Algebra. 320 (2008), 2632-2641. [4] K. Bahmanpour and R. Naghipour, Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra. 321 (2009), 1997-2011. [5] M.P. Brodmann and R.Y. Sharp, Local cohomology; an algebraic introduction with geometric applications, Cambridge University Press, Cambridge, 1998. [6] K. Divaani-Aazar and A. Mafi, Associated primes of local cohomology modules, Proc. Amer. Math. Soc. 133 (2005), 655-660. [7] A. Grothendieck, Local cohomology, Notes by R. Hartshorne, Lecture Notes in Math. 862, Springer, New York, 1966. [8] A. Grothendieck, Cohomologie local des faisceaux coherents et th´ eor´ emes de lefschetz locaux et globaux (SGA2), North-Holland, Amsterdam, 1968. [9] R. Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1970), 145-164. [10] M. Hellus, On the associated primes of a local cohomology module, J. Algebra. 237 (2001), 406419. [11] C. Huneke, Problems on local cohomology, Free resolutions in commutative algebra and algebraic geometry, Res. Notes Math. 2(1992), 93-108. [12] C. Huneke and R.Y. Sharp, Bass numbers of local cohomology module, Trans. Amer. Math. Soc. 339 (1993), 765-779. [13] G. Lyubeznik, Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra), Invent. Math. 113 (1993), 41-55. [14] G. Lyubeznik, Finiteness properties of local cohomology modules for regular local rings of mixed chartacteristic: the unramified case, Comm. Algebra, 28 (2000), 5867-5882. [15] G. Lyubeznik, A partial survey of local cohomology, local cohomology and its applications, Lectures Notes in Pure and Appl. Math. 226 (2002), 121-154. [16] M. Katzman, An example of an infinite set of associated primes of a local cohomology module, J. Algebra 252 (2002), 161-166. [17] K.I. Kawasaki, On the finiteness of Bass numbers of local cohomology modules, Proc. Amer. Math. Soc. 124 (1996), 3275-3279. [18] K.I. Kawasaki, Cofiniteness of local cohomology modules for principal ideals, Bull. London Math. Soc. 30 (1998), 241-246. [19] T. Marley, The associated primes of local cohomology modules over rings of small dimension, Manuscripta Math. 104 (2001), 519-525. [20] T. Marley and J.C. Vassilev, Cofiniteness and associated primes of local cohomology modules, J. Algebra 256 (2002), 180-193. [21] H. Matsumura, Commutative ring theory, Cambridge Univ. Press, Cambridge, UK, 1986. [22] L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra, 285 (2005), 649-668. [23] R. Naghipour and M. Sedghi, A characterization of Cohen-Macaulay modules and local cohomology, Arch. Math. 87 (2006), 303-308.


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[24] P. Schenzel, Proregular sequences, local cohomology and completion, Math. Scand. 92 (2003), 161180. [25] A.K. Singh, P-torsion elements in local cohomology modules, Math. Res. Lett. 7 (2000), 165-176. [26] H. Z¨ oschinger, Minimax modules, J. Algebra, 102 (1986), 1-32. ¨ ber die maximalbedingung f¨ [27] H. Z¨ oschinger, U ur radikalvolle untermoduln, Hokkaido Math. J. 17 (1988), 101-116. Department of Mathematics, Azarbaijan Shahid Madani University, 53714-161, Tabriz, Iran. E-mail address: m [email protected] E-mail address: [email protected] Department of Mathematics, Faculty of Mathematical Sciences, University of Mohaghegh Ardabili, 56199-11367, Ardabil, Iran. E-mail address: [email protected] Department of Mathematics, University of Tabriz, 51666-16471, Tabriz, Iran. E-mail address: [email protected] E-mail address: [email protected]